| Title: | New approach for constructing edge B-spline-like basis functions for $C^1$ and $C^2$ splines over triangulations |
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| Authors: | ID Knez, Marjetka (Author)
ID Šteblaj, Matija (Author) |
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| Files: |  PDF - Presentation file, download (9,42 MB)
MD5: 258BB6075CE01E3ECC0FBF53E8D2A586
 URL - Source URL, visit https://www.sciencedirect.com/science/article/pii/S037704272600556X
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| Language: | English |
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| Typology: | 1.01 - Original Scientific Article |
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| Organization: |  IMFM - Institute of Mathematics, Physics, and Mechanics
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| Abstract: | Splines over triangulations provide a flexible and efficient way to approximate bivariate functions defined over domains, partitioned by triangles. One of the important challenges is the construction of locally supported non-negative basis functions that form a partition of unity, i.e., B-spline-like basis functions. Due to smoothness conditions that depend on the geometry of the triangulation, spline basis functions are typically divided into three groups: triangle, vertex, and edge functions. While the construction of triangle and vertex basis splines is well developed, the computation of non-negative edge basis splines of general degree that form a partition of unity has so far been addressed only for $C^1$ continuous splines, using a completely algebraic approach, with no explicit geometric interpretation ([7]). In this paper a novel approach is presented that extends the control-triangle-based idea known for vertex basis splines and can be geometrically explained and generalized to splines of a higher order of smoothness. The construction is developed on pairs of adjacent triangles forming a strictly convex quadrilateral. Since each smoothness order requires a separate analysis, a detailed study is provided for the $C^1$ and $C^2$ continuous splines of degrees $d \ge 2$ and $d \ge 4$, respectively. Numerical examples are provided to confirm the effectiveness and applicability of the derived construction. |
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| Keywords: | Bernstein-Bézier form, ▫$C^r$▫ splines over triangulations, B-spline-like basis, control triangles, subdivision |
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| Publication status: | Published |
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| Publication version: | Version of Record |
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| Publication date: | 01.01.2027 |
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| Year of publishing: | 2027 |
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| Number of pages: | 22 str. |
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| Numbering: | Vol. 489, article 117914 |
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| PID: | 20.500.12556/DiRROS-30714  |
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| UDC: | 519.6 |
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| ISSN on article: | 0377-0427 |
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| DOI: | 10.1016/j.cam.2026.117914 |
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| COBISS.SI-ID: | 283120643 |
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| Note: |
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| Publication date in DiRROS: | 01.07.2026 |
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| Views: | 163 |
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| Downloads: | 167 |
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